Liquid A decomposes by first-order kinetics, and in a batch reactor \(50 \%\) of \(A\) is converted in a 5 -minute run. How much longer would it take to reach 75\% conversion?

Chemical Reaction Engineering is a core branch of chemical engineering that focuses on the design and optimization of chemical reactors. It's concerned with understanding and utilizing the principles of chemical kinetics and thermodynamics to predict the outcomes of chemical reactions and to design the equipment where these reactions take place. To achieve a designed reaction performance, engineers must balance factors such as concentration, temperature, and pressure, all while ensuring safety and cost-effectiveness.

First-order kinetics is an essential concept within this field. In reactions governed by first-order kinetics, the rate at which a reactant is consumed is directly proportional to its concentration at that instant. This fundamental principle allows engineers to make accurate predictions regarding reaction rates and the time required to complete a reaction under specific conditions, which is crucial for reactor design and scaling up processes from the laboratory to the industrial scale.

The Integrated Rate Law is a mathematical expression that relates the concentrations of reactants to time in a chemical reaction. For first-order kinetics, the integrated rate law is a powerful tool that helps us understand how concentrations change over time. The law is given by the equation \( \ln(\frac{[A]_0}{[A]}) = kt \), where \( [A]_0 \) is the initial concentration of the reactant, \( [A] \) is the concentration at time \( t \), and \( k \) is the first-order rate constant.

In practice, the integrated rate law allows us to calculate the time needed for a reactant to reach a particular concentration, providing insights into the kinetics of the system. Manipulating this equation gives us a direct link between the measurable quantity of concentration and the rate at which that quantity changes—an invaluable relationship for predicting the course of a chemical reaction in a given time frame.

Understanding Reaction Time

Reaction time calculation is a vital aspect of chemical reaction engineering, where you determine how long it will take for a reaction to reach a certain conversion percentage. Specifically, for first-order reactions, the time can be fundamentally related to the concentration change using the integrated rate law. By rearranging the law, we can isolate time (\( t \)) and calculate it when the rate constant (\( k \)) and the desired concentration of reactant are known.

Applying the Concept to Exercises

In our exercise example, knowing 50% conversion occurs in 5 minutes allows us to calculate the rate constant \( k \). With this constant, we can determine how much longer it would take to reach 75% conversion. It's worth noting that the steeper the concentration gradient, the quicker the reaction proceeds. As a higher percentage of reactant is converted, the rate of reaction slows down due to the decrease in reactant concentration. Therefore, reaching from 50% to 75% conversion will take disproportionately longer compared to going from 0% to 50% conversion, which can be counterintuitive and highlights the importance of understanding reaction kinetics for efficient process design and optimization.